Artigos

Esta é uma seleção de artigos sobre investimento usando modelos quantitativos.
Carteiras de Variância Mínima e de Baixa Volatilidade
Carteiras de Variância Mínima no Brasil

Rubesam, Alexandre, and Beltrame, Andre L. (2013), “Carteiras de Variância Mínima no Brasil”, Revista Brasileira de Financas, Vol. 11, No. 1. Available here.

Abstract
Neste trabalho, investigamos carteiras de variância mínima no mercado de ações brasileiro, utilizando diferentes modelos de estimação da matriz de covariância, desde a simples matriz de covariância amostral até modelos GARCH multivariados. Comparamos os resultados das carteiras de variância mínima com os seguintes benchmarks: (i) o índice IBOVESPA, (ii) uma carteira igualmente ponderada, (iii) uma carteira formada através da maximização da razão de Sharpe e (iv) uma carteira formada através da maximização da média geométrica dos retornos. Os resultados mostram que as carteiras de variância mínima apresentam retornos maiores e volatilidades menores do que todos os benchmarks. Também avaliamos o desempenho de carteiras de variância mínima com alavancagem, do tipo 130/30, com resultados análogos. A carteira de variância mínima concentra os investimentos em um número pequeno de ações, com betas baixos em relação ao IBOVESPA, sendo facilmente replicáveis por investidores individuais ou institucionais.
Técnicas Quantitativas de Otimização de Carteiras Aplicadas ao Mercado de Ações Brasileiro

Santos, André A. P., and Tessari, Cristina (2012), “Técnicas Quantitativas de Otimização de Carteiras Aplicadas ao Mercado de Ações Brasileiro”, Revista Brasileira de Financas, Vol. 10, No 3. Available here.

Abstract
Neste artigo examinamos a aplicabilidade e o desempenho fora da amostra das estratégias quantitativas de otimização por média-variância e mínima-variância com relação ao desempenho da carteira ingênua igualmente ponderada (1/N) e da carteira teórica do índice Ibovespa, bem como avaliamos a estabilidade das composições ótimas obtidas. Na obtenção de carteiras ótimas, restritas para venda a descoberto, foram empregadas matrizes de covariâncias estimadas com base em cinco abordagens alternativas: matriz de covariância amostral, matriz RiskMetrics, e três estimadores propostos por Ledoit & Wolf (2003, 2004a,b). Tomando como base diferentes frequências de rebalanceamento das carteiras, as medidas de desempenho fora da amostra indicam que as estratégias quantitativas de otimização proporcionam resultados estatisticamente significativos em termos de menor volatilidade e desempenho ajustado ao risco superior. Além disso, o uso de estimadores mais sofisticados para a matriz de covariâncias gerou carteiras com menor turnover ao longo do tempo.
Low Risk Stocks Outperform within All Observable Markets of the World

Baker, Nardin L., and Haugen, R. A. (2012), “Low Risk Stocks Outperform within All Observable Markets of the World”, Available at SSRN.

Abstract
This article provides global evidence supporting the Low Volatility Anomaly: that low risk stocks consistently provide higher returns than high risk stocks. This study covers 33 different markets during the time period from 1990-2011. (Two previous studies by Haugen & Heins (1972) and Haugen & Baker (1991) show the same negative payoff to risk in time periods 1926-1970 and 1970-1990.) The procedure for our study is intentionally simple, transparent and easily replicable. Our samples include non-survivors. We look at an international universe of stocks beginning with the first month of 1990 until December 2011; we compute the volatility of total return for each company in each country over the previous 24 months. Stocks in each country are ranked by volatility and formed into deciles. In the total universe and in each individual country low risk stocks outperform, the relationship with respect to Sharpe ratios is even more impressive. We believe this anomaly is caused primarily by agency issues, namely the compensation structures and internal stock selection processes at asset management firms which lead institutional investors on average to hold more volatile stocks. The article also addresses the implications for how corporate finance managers make capital investment decision in light of this evidence. The evidence presented here dethrones both CAPM and the Efficient Market Hypothesis.
The Volatility Effect in Emerging Markets

Blitz, David and Van Vliet, Pim (2012), “The Volatility Effect in Emerging Markets”, Available at SSRN.

Abstract
We examine the empirical relation between risk and return in emerging equity markets and find that this relation is flat, or even negative. This is inconsistent with theoretical models such as the CAPM, which predict a positive relation, but consistent with the results of studies for developed equity markets. The volatility effect appears to be growing stronger over time, which we argue might be related to the increased delegated portfolio management in emerging markets. Finally, we find that the volatility effect in emerging markets is only weakly related to that in developed equity markets, which argues against a common-factor explanation.
The Limits to Arbitrage Revisited: The Low-Risk Anomaly

Li, X.; Sullivan, R. and García-Feijóo, L. (2012), “The Limits to Arbitrage Revisited: The Low-Risk Anomaly”, Available at SSRN.

Abstract
We show that over a long study period (1963-2010), the efficacy of trading the well-known low-volatility stock anomaly more limited than widely believed. In particular, extracting excess returns associated with a zero-cost portfolio is meaningfully hampered by high transaction costs reflecting that the abnormal returns are concentrated among low liquidity stocks. Adding to the challenge, the anomalous excess returns quickly reverse requiring traders to rebalance frequently in attempting to extract profits, thus amplifying liquidity needs. Our findings are unchanged for various approaches to measuring the low-volatility anomaly.
The Low Volatility Effect: A Comprehensive Look

Soe, A. M. (2012), “The Low Volatility Effect: A Comprehensive Look, ”, Available at SSRN.

Abstract
We analyze the low volatility effect in the U.S equity market with a focus on the common properties of various low volatility strategies. We examine the two major approaches to constructing low volatility portfolios and apply them to the U.S. equity market: mean-variance optimization-based versus the rankings or quantile-based approaches. Our analysis shows that both approaches are equally effective in reducing portfolio volatility over a long-term investment horizon. We then extend our analysis to the international and emerging markets. Our findings confirm that the low volatility effect is not unique to the U.S. equity markets; it is present on a global scale.
Betting Against Beta

Frazzine, A. and Pedersen, L. H. (2011), “Betting Against Beta”, Available at SSRN.

Abstract
We present a model with leverage and margin constraints that vary across investors and time. We find evidence consistent with each of the model’s five central predictions: (1) Since constrained investors bid up high-beta assets, high beta is associated with low alpha, as we find empirically for U.S. equities, 20 international equity markets, Treasury bonds, corporate bonds, and futures; (2) A betting-against-beta (BAB) factor, which is long leveraged low beta assets and short high-beta assets, produces significant positive risk-adjusted returns; (3) When funding constraints tighten, the return of the BAB factor is low; (4) Increased funding liquidity risk compresses betas toward one; (5) More constrained investors hold riskier assets.
Um Índice de Mínima Variância de Ações Brasileiras

Thomé, César N.; Leal, Ricardo P. C. and Almeida, Vinício de S. (2011), “Um Índice de Mínima Variância de Ações Brasileiras”, Economia Aplicada, Vol 15. Available here.

Abstract
Este trabalho desenvolve um índice de carteiras de mínima variância global (MVP) para as ações mais líquidas do Brasil. Os resultados indicaram que a MVP sem limites sobre os pesos das ações não apresenta diferença significativa de desempenho em relação ao IBOVESPA. A imposição de um peso máximo de dez por cento em cada ação tornou possível superar o IBOVESPA. Contudo, os resultados desta estratégia são comparáveis aos de uma carteira igualmente ponderada e são superados por alguns fundos de gestão ativa em testes fora da amostra. Ainda assim, estas estratégias simples baseadas nestas restrições facilitam a replicação do MVP por investidores individuais e exchange traded funds e sustenta o poder de estratégias ingênuas de investimento.
Benchmarking as Limits to Arbitrage: Understanding the Low-Volatility Anomaly

Baker, M., Bradley, B. and Wurgler, J. (2011), “Benchmarking as Limits to Arbitrage: Understanding the Low-Volatility Anomaly”, Financial Analyst’s Journal, Vol. 67. Available here.

Abstract
Contrary to basic finance principles, high-beta and high-volatility stocks have long underperformed low-beta and low-volatility stocks. This anomaly may be partly explained by the fact that the typical institutional investor’s mandate to beat a fixed benchmark discourages arbitrage activity in both high-alpha, low-beta stocks and low-alpha, high-beta stocks.
Risk and Return in General: Theory and Evidence

Falkenstein, Eric G.(2009), “Risk and Return in General: Theory and Evidence”, Journal of Portfolio Management, pp. 102-113, Fall 2007. Available at SSRN.

Abstract
Empirically, standard, intuitive measures of risk like volatility and beta do not generate a positive correlation with average returns in most asset classes. It is possible that risk, however defined, is not positively related to return as an equilibrium in asset markets. This paper presents a survey of data across 20 different asset classes, and presents a model highlighting the assumptions consistent with no risk premium. The key is that when agents are concerned about relative wealth, risk taking is then deviating from the consensus or market portfolio. In this environment, all risk becomes like idiosyncratic risk in the standard model, avoidable so unpriced.
The Volatility Effect: Lower Risk Without Lower Return

Blitz, David and Van Vliet, Pim (2007), “The Volatility Effect: Lower Risk Without Lower Return”, Journal of Portfolio Management, pp. 102-113, Fall 2007. Available at SSRN.

Abstract
We present empirical evidence that stocks with low volatility earn high risk-adjusted returns. The annual alpha spread of global low versus high volatility decile portfolios amounts to 12% over the 1986-2006 period. We also observe this volatility effect within the US, European and Japanese markets in isolation. Furthermore, we find that the volatility effect cannot be explained by other well-known effects such as value and size. Our results indicate that equity investors overpay for risky stocks. Possible explanations for this phenomenon include (i) leverage restrictions, (ii) inefficient two-step investment processes, and (iii) behavioral biases of private investors. In order to exploit the volatility effect in practice we argue that investors should include low risk stocks as a separate asset class in the strategic asset allocation phase of their investment process.
The Cross-Section of Volatility and Expected Returns

Ang, Andrew; Hodrick, Robert J.; Xing, Yuhang and Zhang, Xiaoyan (2006), “The Cross-Section of Volatility and Expected Returns”, Journal of Finance, 61(1), 259-299. Available at SSRN.

Abstract
We examine how volatility risk, both at the aggregate market and individual stock level, is priced in the cross-section of expected stock returns. Stocks that have past high sensitivities to innovations in aggregate volatility have low average returns. We also find that stocks with past high idiosyncratic volatility have abysmally low returns, but this cannot be explained by exposure to aggregate volatility risk. The low returns earned by stocks with high exposure to systematic volatility risk and the low returns of stocks with high idiosyncratic volatility cannot be explained by the standard size, book-to-market, or momentum effects, and are not subsumed by liquidity or volume effects.
Minimum-Variance Portfolios in the US Equity Market

Clarke, Roger; Harindra de Silva and Throley, Steven (2006), “Minimum-Variance Portfolios in the US Equity Market”, Journal of Portfolio Management,10-24, 2006. Available at here.

Otimização de Carteiras
Leverage Aversion and Risk Parity

Asness, C.; Frazzini, A. and Pedersen, L. H. (2012), “Leverage Aversion and Risk Parity”, Financial Analysts Journal, Vol. 68, N. 1. Available here.

Abstract
The authors show that leverage aversion changes the predictions of modern portfolio theory: Safer assets must offer higher risk-adjusted returns than riskier assets. Consuming the high risk-adjusted returns of safer assets requires leverage, creating an opportunity for investors with the ability to apply leverage. Risk parity portfolios exploit this opportunity by equalizing the risk allocation across asset classes, thus overweighting safer assets relative to their weight in the market portfolio.
Properties of the Most Diversified Portfolio

Choueifaty, Y.; Froidure, T. and Reynier, J. (2011), “Properties of the Most Diversified Portfolio”, Journal of Investment Strategies, Vol. 2, Available at SSRN.

Abstract
This article expands upon “Toward Maximum Diversification” by Choueifaty and Coignard [2008]. We present new mathematical properties of the Diversification Ratio and Most Diversified Portfolio (MDP), and investigate the optimality of the MDP in a mean-variance framework. We also introduce a set of “Portfolio Invariance Properties,” providing the basic rules an unbiased portfolio construction process should respect. The MDP is then compared in light of these rules to popular methodologies (equal weights, equal risk contribution, minimum variance), and their performance is investigated over the past decade, using the MSCI World as reference universe. We believe that the results obtained in this article show that the MDP is a strong candidate for being the un-diversifiable portfolio, and as such delivers investors with the full benefit of the equity premium.
Risk Parity, Maximum Diversification, and Minimum Variance: An Analytic Perspective

Carke, R.; De Silva, H. and Thorley, S. (2011), “Risk Parity, Maximum Diversification, and Minimum Variance: An Analytic Perspective”, Available at SSRN.

Abstract
Analytic solutions to Risk Parity, Maximum Diversification, and Minimum Variance portfolios provide useful perspectives about their construction and composition. Individual asset weights depend on both systematic and idiosyncratic risk in all three risk-based portfolios, but systematic risk eliminates many investable assets in long-only constrained Maximum Diversification and Minimum Variance portfolios. On the other hand, all investable assets are included in Risk Parity portfolios, and idiosyncratic risk has little impact on the magnitude of the weights. The algebraic forms for optimal asset weights derived in this paper yield generalizable properties of risk-based portfolios, in contrast to empirical simulations that employ a specific set of historical returns, proprietary risk models, and multiple constraints. The analytic solutions reveal precisely how the various kinds of predicted risk impact the relative magnitude of security weights under each type of risk-based portfolio construction.
Risk Parity Portfolio vs. Other Asset Allocation Heuristic Portfolios

Chave, D. B.; Hsu, J.; Li, F. and Shakernia, O. (2010), “Risk Parity Portfolio vs. Other Asset Allocation Heuristic Portfolios,” Available at SSRN.

Abstract
In this paper, we conduct a horse race between representative Risk Parity portfolios and other asset allocation strategies, including equal weighting, minimum-variance, mean-variance optimization, and the classic 60/40 equity/bond portfolio. We find that the traditional Risk Parity portfolio construction does not consistently outperform on a risk-adjusted basis the equal weighting or a model pension fund portfolio anchored to the 60/40 equity/bond portfolio structure. However, it does significantly outperform optimized allocation strategies such as minimum-variance and mean-variance efficient portfolios on a consistent basis. Over the past 30 years, the Sharpe Ratios of the Risk Parity and the equal weighting portfolio have been much more stable across decade-long sub-periods than either the 60/40 portfolio or the optimized portfolios. Although Risk Parity performs on par with equal weighting, it does provide better diversification in terms of risk allocation and, thus, warrants further consideration as an asset allocation strategy. However, we show that the Risk Parity strategy’s performance can be highly dependent on the investment universe. Thus, to execute on Risk Parity successfully, careful selection of asset classes is very critical and, for the time being, remains an art rather than a science.
On the Properties of Equally-Weighted Risk Contributions Portfolios

Maillard, S.; Roncalli, T. and Teiletche, J. (2008), “On the Properties of Equally-Weighted Risk Contributions Portfolios”, Available at SSRN.

Abstract
Minimum variance and equally-weighted portfolios have recently prompted great interest both from academic researchers and market practitioners, as their construction does not rely on expected average returns and is therefore assumed to be robust. In this paper, we consider a related approach, where the risk contribution from each portfolio components is made equal, which maximizes diversification of risk (at least on an ex-ante basis). Roughly speaking, the resulting portfolio is similar to a minimum variance portfolio subject to a diversification constraint on the weights of its components. We derive the theoretical properties of such a portfolio and show that its volatility is located between those of minimum variance and equally-weighted portfolios. Empirical applications confirm that ranking. All in all, equally-weighted risk contributions portfolios appear to be an attractive alternative to minimum variance and equally-weighted portfolios and might be considered a good trade-off between those two approaches in terms of absolute level of risk, risk budgeting and diversification.
Operações com Pares e Arbitragem Estatística
Seleção de uma Carteira de Pares de Ações Usando Cointegração: Uma Estrategia de Arbitragem Estatística

Caldeira, J. F. and Moura, G. V. (2013), “Seleção de uma Carteira de Pares de Ações Usando Cointegração: Uma Estrategia de Arbitragem Estatística”, Revista Brasileira de Finanças, Vol. 11, N. 1, Available here.

Abstract
Estratégias de arbitragem estatística como pairs trading e suas generalizações dependem da construção de spreads estacionários com certo grau de previsibilidade. Este artigo aplica testes de cointegração para identificar ativos para serem usados em estratégias de pairs trading. Além de estimar o equilíbrio de longo prazo e de modelar os resíduos resultantes, pares de ações são selecionados baseados em um indicador de lucratividade para compor um portfólio de pares. O retorno da estratégia é avaliado com dados diários da Bovespa durante o período de janeiro de 2005 até outubro de 2012. A análise empírica mostra que a estratégia proposta obtém excessos de retorno da ordem de 16.38% ao ano, índice de Sharpe de 1.34 e baixa correlação com o Ibovespa.
Estratégia Long-Short, Neutra ao Mercado, e Index Tracking Baseadas em Portfólios Cointegrados

Caldeira, J. F. and Portugal, M. S. (2010), “Estratégia Long-Short, Neutra ao Mercado, e Index Tracking Baseadas em Portfólios Cointegrados”, Revista Brasileira de Finanças, Vol. 8, N. 4, Available here.

Abstract
Modelos de otimização de carteiras baseados na análise média-variância apresentam dificuldadespara estimação das matrizes de covariância, o que leva a necessidade de métodos ad hoc para limitar ou suavizar as alocações eficientes recomendadas pelo modelo. Embora as carteiras resultantes sejam eficientes, não é assegurado que o tracking error seja estacionário, podendo a carteira se distanciar do benchmark, exigindo frequentes recomposições. Este artigo aplica metodologia de cointegração para otimização de carteiras que são utilizadas em estratégias index tracking e long-short. As carteiras resultantes apresentam elevada estabilidade, refletindo em baixos custos de ajuste. Níveis de retorno e volatilidade superiores aos benchmarks mostram que a metodologia é uma ferramenta eficiente e capaz de gerar resultados robustos, se caracterizando como uma atraente ferramenta para a gestão quantitativa de recursos.
Arbitragem Estatística e Estratégia Long-Short Pairs Trading, Abordagem de Cointegração Aplicada a Dados do Mercado Brasileiro

Caldeira, J. F. (2010), “Arbitragem Estatística e Estratégia Long-Short Pairs Trading, Abordagem de Cointegração Aplicada a Dados do Mercado Brasileiro”, Available here.

Abstract
A motivação para este artigo é aplicar os testes de cointegração de Johansen e Engle-Granger para identificar pares de ações a serem usados numa estratégia de pairs trading. Estratégias pairs trading são um tipo de arbitragem estatística de valor relativo que busca explorar desvios temporários de relações de equilíbrio de longo prazo entre pares de ações. Além de estimar uma relação de equilíbrio de longo prazo para identificar os pares e modelar os resíduos resultantes, com característica de reversão à média, empregamos um indicador de rentabilidade em simulações dentro da amostra para selecionar pares de ações para compor uma carteira pairs trading em testes fora da amostra. Nós analisamos a rentabilidade da estratégia de investimento pairs trading com dados diários do mercado de ações brasileiro no período de janeiro de 2005 a dezembro de 2009. Aplicando regras de trading simples obtivemos rentabilidade média de 17;49% ao ano para uma carteira de pares de ações que se auto financia. Apresentando índice de Sharpe de 1:29 e baixa correlação com o mercado, os resultados reforçam o uso do conceito de cointegração na gestão quantitativa de fundos.
Statistical Arbitrage in the U.S. Equities Market

Avellaneda, M. and Lee, J. (2008), “Statistical Arbitrage in the U.S. Equities Market”, Available at SSRN

Abstract
We study model-driven statistical arbitrage strategies in U.S. equities. Trading signals are generated in two ways: using Principal Component Analysis and using sector ETFs. In both cases, we consider the residuals, or idiosyncratic components of stock returns, and model them as a mean-reverting process, which leads naturally to “contrarian” trading signals. The main contribution of the paper is the back-testing and comparison of market-neutral PCA- and ETF- based strategies over the broad universe of U.S. equities. Back-testing shows that, after accounting for transaction costs, PCA-based strategies have an average annual Sharpe ratio of 1.44 over the period 1997 to 2007, with a much stronger performances prior to 2003: during 2003-2007, the average Sharpe ratio of PCA-based strategies was only 0.9. On the other hand, strategies based on ETFs achieved a Sharpe ratio of 1.1 from 1997 to 2007, but experience a similar degradation of performance after 2002. We introduce a method to take into account daily trading volume information in the signals (using “trading time” as opposed to calendar time), and observe significant improvements in performance in the case of ETF-based signals. ETF strategies which use volume information achieve a Sharpe ratio of 1.51 from 2003 to 2007. The paper also relates the performance of mean-reversion statistical arbitrage strategies with the stock market cycle. In particular, we study in some detail the performance of the strategies during the liquidity crisis of the summer of 2007. We obtain results which are consistent with Khandani and Lo (2007) and validate their “unwinding” theory for the quant fund drawndown of August 2007.
Pairs Trading: Performance of a Relative Value Arbitrage Rule

Gatev, Evan, Goetzmann, William N. and Rouwenhorst, K. Geert (2006), “Pairs Trading: Performance of a Relative Value Arbitrage Rule”, Yale ICF Working Paper No. 08-03. Available at SSRN

Abstract
We test a Wall Street investment strategy, pairs trading, with daily data over 1962-2002. Stocks are matched into pairs with minimum distance between normalized historical prices. A simple trading rule yields average annualized excess returns of up to 11 percent for selffinancing portfolios of pairs. The profits typically exceed conservative transaction costs estimates. Bootstrap results suggest that the pairs effect differs from previously-documented reversal profits. Robustness of the excess returns indicates that pairs trading profits from temporary mis-pricing of close substitutes. We link the profitability to the presence of a common factor in the returns, different from conventional risk measures.
Evaluation of Pairs Trading Strategy at the Brazilian Financial Market

Perlin, M. (2006), “Evaluation of Pairs Trading Strategy at the Brazilian Financial Market”, Journal of Derivatives & Hedge Funds, Vol. 15. Available at SSRN

Abstract
Pairs trading is a popular trading strategy that tries to take advantage of market inefficiencies in order to obtain profit. The idea is simple: find two stocks that move together and take long/short positions when they diverge abnormally, hoping that the prices will converge in the future. From the academic point of view of weak market efficiency theory, pairs trading strategy shouldn’t present positive performance since, according to it, the actual price of a stock reflects its past trading data, including historical prices. This leaves us with a question, does pairs trading strategy presents positive performance for the Brazilian market? The main objective of this research is to verify the performance and risk of pairs trading in the Brazilian financial market for different frequencies of the database, daily, weekly and monthly prices for the same time period. The main conclusion of this simulation is that pairs trading strategy was a profitable and market neutral strategy at the Brazilian Market. Such profitability was consistent over a region of the strategy’s parameters. The best results were found for the highest frequency (daily), which is an intuitive result.
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